In simple terms, how would you explain (perhaps with simple examples) the difference between fixed effect, random effect and mixed effect models?
Mixed Models – Difference Between Fixed Effect and Random Effect
definitionfixed-effects-modelmixed modelrandom-effects-model
Related Solutions
This seems a great question as it touches a nomenclature issue in econometrics that disturbs students when switching to statistic literature (books, teachers, etc). I suggest you http://www.amazon.com/Econometric-Analysis-Cross-Section-Panel/dp/0262232197 chapter 10.
Assume that your variable of interest $y_{it}$ is observed in two dimensions (e.g. individuals and time) depends on observed characteristics $x_{it}$ and unobserved ones $u_{it}$. If $y_{it}$ are observed wages then we may argue that it's determined by observed (education) and unobserved skills (talents, etc.). But it's clear that unobserved skills may be correlated with educational levels. So that leads to the error decomposition: $u_{it} = e_{it}+v_i$ where $v_i$ is the error (random) component that we may assume to be correlated with the $x$'s. i.e. $v_i$ models the individual's unobserved skills as a random individual component.
Thus the model becomes:
$y_{it} = \sum_j\theta_jx_j + e_{it}+ v_{i} $
This model is usually labeled as a FE model, but as Wooldridge argues it would be wiser to call it a RE model with correlated error component whereas if $v_i$ is not correlated to the $x's$ it becomes a RE model. So this answer your second question, the FE setup is more general as it allows for correlation between $v_i$ and the $x's$.
Older books in econometrics tend to refer to FE to a model with individual specific constants, unfortunately this is still present in nowadays literature (I guess that in statistics they never have had this confussion. I definitevely suggest the Wooldridge lectures that develops the potential missunderstanding issue)
Whether a coefficient from a model has a causal interpretation mostly depends on the other variables included or the way that unobserved but relevant variables are controlled for. For example, in an earnings regression of the type $$\ln(y_{i}) = \alpha + \delta S_{i} + \gamma A_{i} + X'\beta + \epsilon$$ where the dependent variable is log earnings, $S_{i}$ is years of education, $A_{i}$ is ability and $X$ are other relevant variables that affect wages like parental background, age, gender, etc.
Assume $A_{i}$ and $S_{i}$ are correlated and that there are no other endogeneity issues or measurement error. If you can observe $S_{i}$, $A_{i}$ and $X$, then the coefficient $\delta$ has a causal interpretation, i.e. it is the causal effect of an additional year of education on earnings - holding all else constant. This ceteris paribus assumption is what makes causality.
To extend this example to your fixed effects model, if you have panel data and you don't observe $A_{i}$, you can still consistently estimate $\delta$ using fixed effects. Suppose $S_{i}$ varies over time and $A_{i}$ does not vary over time, then $$\ln(y_{i}) = \eta + \delta S_{i} + X'\beta + \epsilon$$ the absorbing variable $\eta = \alpha + A_{i} + G_{i}$ includes all observed and unobserved variables that do not vary over time, like the intercept or $G_{i} =$ gender, place of birth, etc. So it pulls $A_{i}$ out of the error and hence removes the endogeneity problem (remember $A_{i}$ and $S_{i}$ are correlated, so if $A_{i}$ is in the error, $S_{i}$ will be correlated with the error). The problem is that $A_{i}$ is likely not to be fixed over time as for instance mental capabilities and productivity diminish with old age.
In theory, I could go on providing examples for each type of your models but I guess you get the idea. Whether or not you estimate a causal effect depends on the included (and omitted!) variables AND on the assumptions of the model. So see what kind of data you have at hand, what you can control for in terms of relevant variables for the relationship you are after (perhaps you don't even have an endogeneity problem), and what assumptions are the most realistic for your analysis to be credible. If you want to dig a little deeper into the topic of causal effects estimation, Mostly Harmless Econometrics by Angrist and Pischke is an excellent book. Otherwise you will find plenty of lecture notes online.
Best Answer
Statistician Andrew Gelman says that the terms 'fixed effect' and 'random effect' have variable meanings depending on who uses them. Perhaps you can pick out which one of the 5 definitions applies to your case. In general it may be better to either look for equations which describe the probability model the authors are using (when reading) or write out the full probability model you want to use (when writing).